Given triangle $ABC$. A circle passes through $B$ and $C$ and intersects sides $AB$ and $AC$ at points $C_1$ and $B_1$ respectively. The line $B_1C_1$ intersects the circle $\omega$, which is the circumcircle of $ABC$, at points $X$ and $Y$. Lines $BB_1$ and $CC_1$ intersect $\omega$ at points $P$ and $Q$ respectively ($P \neq B$ and $Q \neq C$). Prove that $QX=PY$.